The rewrite relation of the following TRS is considered.
active(terms(N)) |
→ |
mark(cons(recip(sqr(N)),terms(s(N)))) |
(1) |
active(sqr(0)) |
→ |
mark(0) |
(2) |
active(sqr(s(X))) |
→ |
mark(s(add(sqr(X),dbl(X)))) |
(3) |
active(dbl(0)) |
→ |
mark(0) |
(4) |
active(dbl(s(X))) |
→ |
mark(s(s(dbl(X)))) |
(5) |
active(add(0,X)) |
→ |
mark(X) |
(6) |
active(add(s(X),Y)) |
→ |
mark(s(add(X,Y))) |
(7) |
active(first(0,X)) |
→ |
mark(nil) |
(8) |
active(first(s(X),cons(Y,Z))) |
→ |
mark(cons(Y,first(X,Z))) |
(9) |
active(half(0)) |
→ |
mark(0) |
(10) |
active(half(s(0))) |
→ |
mark(0) |
(11) |
active(half(s(s(X)))) |
→ |
mark(s(half(X))) |
(12) |
active(half(dbl(X))) |
→ |
mark(X) |
(13) |
mark(terms(X)) |
→ |
active(terms(mark(X))) |
(14) |
mark(cons(X1,X2)) |
→ |
active(cons(mark(X1),X2)) |
(15) |
mark(recip(X)) |
→ |
active(recip(mark(X))) |
(16) |
mark(sqr(X)) |
→ |
active(sqr(mark(X))) |
(17) |
mark(s(X)) |
→ |
active(s(mark(X))) |
(18) |
mark(0) |
→ |
active(0) |
(19) |
mark(add(X1,X2)) |
→ |
active(add(mark(X1),mark(X2))) |
(20) |
mark(dbl(X)) |
→ |
active(dbl(mark(X))) |
(21) |
mark(first(X1,X2)) |
→ |
active(first(mark(X1),mark(X2))) |
(22) |
mark(nil) |
→ |
active(nil) |
(23) |
mark(half(X)) |
→ |
active(half(mark(X))) |
(24) |
terms(mark(X)) |
→ |
terms(X) |
(25) |
terms(active(X)) |
→ |
terms(X) |
(26) |
cons(mark(X1),X2) |
→ |
cons(X1,X2) |
(27) |
cons(X1,mark(X2)) |
→ |
cons(X1,X2) |
(28) |
cons(active(X1),X2) |
→ |
cons(X1,X2) |
(29) |
cons(X1,active(X2)) |
→ |
cons(X1,X2) |
(30) |
recip(mark(X)) |
→ |
recip(X) |
(31) |
recip(active(X)) |
→ |
recip(X) |
(32) |
sqr(mark(X)) |
→ |
sqr(X) |
(33) |
sqr(active(X)) |
→ |
sqr(X) |
(34) |
s(mark(X)) |
→ |
s(X) |
(35) |
s(active(X)) |
→ |
s(X) |
(36) |
add(mark(X1),X2) |
→ |
add(X1,X2) |
(37) |
add(X1,mark(X2)) |
→ |
add(X1,X2) |
(38) |
add(active(X1),X2) |
→ |
add(X1,X2) |
(39) |
add(X1,active(X2)) |
→ |
add(X1,X2) |
(40) |
dbl(mark(X)) |
→ |
dbl(X) |
(41) |
dbl(active(X)) |
→ |
dbl(X) |
(42) |
first(mark(X1),X2) |
→ |
first(X1,X2) |
(43) |
first(X1,mark(X2)) |
→ |
first(X1,X2) |
(44) |
first(active(X1),X2) |
→ |
first(X1,X2) |
(45) |
first(X1,active(X2)) |
→ |
first(X1,X2) |
(46) |
half(mark(X)) |
→ |
half(X) |
(47) |
half(active(X)) |
→ |
half(X) |
(48) |
active#(terms(N)) |
→ |
mark#(cons(recip(sqr(N)),terms(s(N)))) |
(49) |
active#(terms(N)) |
→ |
cons#(recip(sqr(N)),terms(s(N))) |
(50) |
active#(terms(N)) |
→ |
recip#(sqr(N)) |
(51) |
active#(terms(N)) |
→ |
sqr#(N) |
(52) |
active#(terms(N)) |
→ |
terms#(s(N)) |
(53) |
active#(terms(N)) |
→ |
s#(N) |
(54) |
active#(sqr(0)) |
→ |
mark#(0) |
(55) |
active#(sqr(s(X))) |
→ |
mark#(s(add(sqr(X),dbl(X)))) |
(56) |
active#(sqr(s(X))) |
→ |
s#(add(sqr(X),dbl(X))) |
(57) |
active#(sqr(s(X))) |
→ |
add#(sqr(X),dbl(X)) |
(58) |
active#(sqr(s(X))) |
→ |
sqr#(X) |
(59) |
active#(sqr(s(X))) |
→ |
dbl#(X) |
(60) |
active#(dbl(0)) |
→ |
mark#(0) |
(61) |
active#(dbl(s(X))) |
→ |
mark#(s(s(dbl(X)))) |
(62) |
active#(dbl(s(X))) |
→ |
s#(s(dbl(X))) |
(63) |
active#(dbl(s(X))) |
→ |
s#(dbl(X)) |
(64) |
active#(dbl(s(X))) |
→ |
dbl#(X) |
(65) |
active#(add(0,X)) |
→ |
mark#(X) |
(66) |
active#(add(s(X),Y)) |
→ |
mark#(s(add(X,Y))) |
(67) |
active#(add(s(X),Y)) |
→ |
s#(add(X,Y)) |
(68) |
active#(add(s(X),Y)) |
→ |
add#(X,Y) |
(69) |
active#(first(0,X)) |
→ |
mark#(nil) |
(70) |
active#(first(s(X),cons(Y,Z))) |
→ |
mark#(cons(Y,first(X,Z))) |
(71) |
active#(first(s(X),cons(Y,Z))) |
→ |
cons#(Y,first(X,Z)) |
(72) |
active#(first(s(X),cons(Y,Z))) |
→ |
first#(X,Z) |
(73) |
active#(half(0)) |
→ |
mark#(0) |
(74) |
active#(half(s(0))) |
→ |
mark#(0) |
(75) |
active#(half(s(s(X)))) |
→ |
mark#(s(half(X))) |
(76) |
active#(half(s(s(X)))) |
→ |
s#(half(X)) |
(77) |
active#(half(s(s(X)))) |
→ |
half#(X) |
(78) |
active#(half(dbl(X))) |
→ |
mark#(X) |
(79) |
mark#(terms(X)) |
→ |
active#(terms(mark(X))) |
(80) |
mark#(terms(X)) |
→ |
terms#(mark(X)) |
(81) |
mark#(terms(X)) |
→ |
mark#(X) |
(82) |
mark#(cons(X1,X2)) |
→ |
active#(cons(mark(X1),X2)) |
(83) |
mark#(cons(X1,X2)) |
→ |
cons#(mark(X1),X2) |
(84) |
mark#(cons(X1,X2)) |
→ |
mark#(X1) |
(85) |
mark#(recip(X)) |
→ |
active#(recip(mark(X))) |
(86) |
mark#(recip(X)) |
→ |
recip#(mark(X)) |
(87) |
mark#(recip(X)) |
→ |
mark#(X) |
(88) |
mark#(sqr(X)) |
→ |
active#(sqr(mark(X))) |
(89) |
mark#(sqr(X)) |
→ |
sqr#(mark(X)) |
(90) |
mark#(sqr(X)) |
→ |
mark#(X) |
(91) |
mark#(s(X)) |
→ |
active#(s(mark(X))) |
(92) |
mark#(s(X)) |
→ |
s#(mark(X)) |
(93) |
mark#(s(X)) |
→ |
mark#(X) |
(94) |
mark#(0) |
→ |
active#(0) |
(95) |
mark#(add(X1,X2)) |
→ |
active#(add(mark(X1),mark(X2))) |
(96) |
mark#(add(X1,X2)) |
→ |
add#(mark(X1),mark(X2)) |
(97) |
mark#(add(X1,X2)) |
→ |
mark#(X1) |
(98) |
mark#(add(X1,X2)) |
→ |
mark#(X2) |
(99) |
mark#(dbl(X)) |
→ |
active#(dbl(mark(X))) |
(100) |
mark#(dbl(X)) |
→ |
dbl#(mark(X)) |
(101) |
mark#(dbl(X)) |
→ |
mark#(X) |
(102) |
mark#(first(X1,X2)) |
→ |
active#(first(mark(X1),mark(X2))) |
(103) |
mark#(first(X1,X2)) |
→ |
first#(mark(X1),mark(X2)) |
(104) |
mark#(first(X1,X2)) |
→ |
mark#(X1) |
(105) |
mark#(first(X1,X2)) |
→ |
mark#(X2) |
(106) |
mark#(nil) |
→ |
active#(nil) |
(107) |
mark#(half(X)) |
→ |
active#(half(mark(X))) |
(108) |
mark#(half(X)) |
→ |
half#(mark(X)) |
(109) |
mark#(half(X)) |
→ |
mark#(X) |
(110) |
terms#(mark(X)) |
→ |
terms#(X) |
(111) |
terms#(active(X)) |
→ |
terms#(X) |
(112) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(113) |
cons#(X1,mark(X2)) |
→ |
cons#(X1,X2) |
(114) |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(115) |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(116) |
recip#(mark(X)) |
→ |
recip#(X) |
(117) |
recip#(active(X)) |
→ |
recip#(X) |
(118) |
sqr#(mark(X)) |
→ |
sqr#(X) |
(119) |
sqr#(active(X)) |
→ |
sqr#(X) |
(120) |
s#(mark(X)) |
→ |
s#(X) |
(121) |
s#(active(X)) |
→ |
s#(X) |
(122) |
add#(mark(X1),X2) |
→ |
add#(X1,X2) |
(123) |
add#(X1,mark(X2)) |
→ |
add#(X1,X2) |
(124) |
add#(active(X1),X2) |
→ |
add#(X1,X2) |
(125) |
add#(X1,active(X2)) |
→ |
add#(X1,X2) |
(126) |
dbl#(mark(X)) |
→ |
dbl#(X) |
(127) |
dbl#(active(X)) |
→ |
dbl#(X) |
(128) |
first#(mark(X1),X2) |
→ |
first#(X1,X2) |
(129) |
first#(X1,mark(X2)) |
→ |
first#(X1,X2) |
(130) |
first#(active(X1),X2) |
→ |
first#(X1,X2) |
(131) |
first#(X1,active(X2)) |
→ |
first#(X1,X2) |
(132) |
half#(mark(X)) |
→ |
half#(X) |
(133) |
half#(active(X)) |
→ |
half#(X) |
(134) |
The dependency pairs are split into 10
components.
-
The
1st
component contains the
pair
mark#(terms(X)) |
→ |
active#(terms(mark(X))) |
(80) |
active#(terms(N)) |
→ |
mark#(cons(recip(sqr(N)),terms(s(N)))) |
(49) |
mark#(terms(X)) |
→ |
mark#(X) |
(82) |
mark#(cons(X1,X2)) |
→ |
active#(cons(mark(X1),X2)) |
(83) |
active#(sqr(s(X))) |
→ |
mark#(s(add(sqr(X),dbl(X)))) |
(56) |
mark#(cons(X1,X2)) |
→ |
mark#(X1) |
(85) |
mark#(recip(X)) |
→ |
active#(recip(mark(X))) |
(86) |
active#(dbl(s(X))) |
→ |
mark#(s(s(dbl(X)))) |
(62) |
mark#(recip(X)) |
→ |
mark#(X) |
(88) |
mark#(sqr(X)) |
→ |
active#(sqr(mark(X))) |
(89) |
active#(add(0,X)) |
→ |
mark#(X) |
(66) |
mark#(sqr(X)) |
→ |
mark#(X) |
(91) |
mark#(s(X)) |
→ |
active#(s(mark(X))) |
(92) |
active#(add(s(X),Y)) |
→ |
mark#(s(add(X,Y))) |
(67) |
mark#(s(X)) |
→ |
mark#(X) |
(94) |
mark#(add(X1,X2)) |
→ |
active#(add(mark(X1),mark(X2))) |
(96) |
active#(first(s(X),cons(Y,Z))) |
→ |
mark#(cons(Y,first(X,Z))) |
(71) |
mark#(add(X1,X2)) |
→ |
mark#(X1) |
(98) |
mark#(add(X1,X2)) |
→ |
mark#(X2) |
(99) |
mark#(dbl(X)) |
→ |
active#(dbl(mark(X))) |
(100) |
active#(half(s(s(X)))) |
→ |
mark#(s(half(X))) |
(76) |
mark#(dbl(X)) |
→ |
mark#(X) |
(102) |
mark#(first(X1,X2)) |
→ |
active#(first(mark(X1),mark(X2))) |
(103) |
active#(half(dbl(X))) |
→ |
mark#(X) |
(79) |
mark#(first(X1,X2)) |
→ |
mark#(X1) |
(105) |
mark#(first(X1,X2)) |
→ |
mark#(X2) |
(106) |
mark#(half(X)) |
→ |
active#(half(mark(X))) |
(108) |
mark#(half(X)) |
→ |
mark#(X) |
(110) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active#(x1)] |
= |
-2 + 2 · x1
|
[add(x1, x2)] |
= |
2 |
[cons(x1, x2)] |
= |
-2 |
[dbl(x1)] |
= |
2 |
[first(x1, x2)] |
= |
2 |
[half(x1)] |
= |
2 |
[recip(x1)] |
= |
0 |
[s(x1)] |
= |
-2 |
[sqr(x1)] |
= |
2 |
[terms(x1)] |
= |
2 |
[mark(x1)] |
= |
-2 |
[active(x1)] |
= |
2 |
[0] |
= |
0 |
[nil] |
= |
0 |
[mark#(x1)] |
= |
2 |
together with the usable
rules
terms(active(X)) |
→ |
terms(X) |
(26) |
terms(mark(X)) |
→ |
terms(X) |
(25) |
sqr(active(X)) |
→ |
sqr(X) |
(34) |
sqr(mark(X)) |
→ |
sqr(X) |
(33) |
recip(active(X)) |
→ |
recip(X) |
(32) |
recip(mark(X)) |
→ |
recip(X) |
(31) |
s(active(X)) |
→ |
s(X) |
(36) |
s(mark(X)) |
→ |
s(X) |
(35) |
cons(X1,mark(X2)) |
→ |
cons(X1,X2) |
(28) |
cons(mark(X1),X2) |
→ |
cons(X1,X2) |
(27) |
cons(active(X1),X2) |
→ |
cons(X1,X2) |
(29) |
cons(X1,active(X2)) |
→ |
cons(X1,X2) |
(30) |
dbl(active(X)) |
→ |
dbl(X) |
(42) |
dbl(mark(X)) |
→ |
dbl(X) |
(41) |
add(X1,mark(X2)) |
→ |
add(X1,X2) |
(38) |
add(mark(X1),X2) |
→ |
add(X1,X2) |
(37) |
add(active(X1),X2) |
→ |
add(X1,X2) |
(39) |
add(X1,active(X2)) |
→ |
add(X1,X2) |
(40) |
first(X1,mark(X2)) |
→ |
first(X1,X2) |
(44) |
first(mark(X1),X2) |
→ |
first(X1,X2) |
(43) |
first(active(X1),X2) |
→ |
first(X1,X2) |
(45) |
first(X1,active(X2)) |
→ |
first(X1,X2) |
(46) |
half(active(X)) |
→ |
half(X) |
(48) |
half(mark(X)) |
→ |
half(X) |
(47) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
mark#(cons(X1,X2)) |
→ |
active#(cons(mark(X1),X2)) |
(83) |
mark#(recip(X)) |
→ |
active#(recip(mark(X))) |
(86) |
mark#(s(X)) |
→ |
active#(s(mark(X))) |
(92) |
could be deleted.
1.1.1.1 Reduction Pair Processor
Using the
prec(mark#) |
= |
1 |
|
stat(mark#) |
= |
mul
|
prec(terms) |
= |
5 |
|
stat(terms) |
= |
mul
|
prec(active#) |
= |
0 |
|
stat(active#) |
= |
mul
|
prec(sqr) |
= |
4 |
|
stat(sqr) |
= |
mul
|
prec(s) |
= |
1 |
|
stat(s) |
= |
mul
|
prec(add) |
= |
2 |
|
stat(add) |
= |
mul
|
prec(dbl) |
= |
3 |
|
stat(dbl) |
= |
mul
|
prec(0) |
= |
6 |
|
stat(0) |
= |
mul
|
prec(first) |
= |
7 |
|
stat(first) |
= |
mul
|
prec(nil) |
= |
6 |
|
stat(nil) |
= |
mul
|
π(mark#) |
= |
[1] |
π(terms) |
= |
[1] |
π(active#) |
= |
[1] |
π(mark) |
= |
1 |
π(cons) |
= |
1 |
π(recip) |
= |
1 |
π(sqr) |
= |
[1] |
π(s) |
= |
[1] |
π(add) |
= |
[1,2] |
π(dbl) |
= |
[1] |
π(0) |
= |
[] |
π(first) |
= |
[1,2] |
π(half) |
= |
1 |
π(active) |
= |
1 |
π(nil) |
= |
[] |
the
pairs
mark#(terms(X)) |
→ |
active#(terms(mark(X))) |
(80) |
active#(terms(N)) |
→ |
mark#(cons(recip(sqr(N)),terms(s(N)))) |
(49) |
mark#(terms(X)) |
→ |
mark#(X) |
(82) |
active#(sqr(s(X))) |
→ |
mark#(s(add(sqr(X),dbl(X)))) |
(56) |
active#(dbl(s(X))) |
→ |
mark#(s(s(dbl(X)))) |
(62) |
mark#(sqr(X)) |
→ |
active#(sqr(mark(X))) |
(89) |
active#(add(0,X)) |
→ |
mark#(X) |
(66) |
mark#(sqr(X)) |
→ |
mark#(X) |
(91) |
active#(add(s(X),Y)) |
→ |
mark#(s(add(X,Y))) |
(67) |
mark#(s(X)) |
→ |
mark#(X) |
(94) |
mark#(add(X1,X2)) |
→ |
active#(add(mark(X1),mark(X2))) |
(96) |
active#(first(s(X),cons(Y,Z))) |
→ |
mark#(cons(Y,first(X,Z))) |
(71) |
mark#(add(X1,X2)) |
→ |
mark#(X1) |
(98) |
mark#(add(X1,X2)) |
→ |
mark#(X2) |
(99) |
mark#(dbl(X)) |
→ |
active#(dbl(mark(X))) |
(100) |
active#(half(s(s(X)))) |
→ |
mark#(s(half(X))) |
(76) |
mark#(dbl(X)) |
→ |
mark#(X) |
(102) |
mark#(first(X1,X2)) |
→ |
active#(first(mark(X1),mark(X2))) |
(103) |
active#(half(dbl(X))) |
→ |
mark#(X) |
(79) |
mark#(first(X1,X2)) |
→ |
mark#(X1) |
(105) |
mark#(first(X1,X2)) |
→ |
mark#(X2) |
(106) |
mark#(half(X)) |
→ |
active#(half(mark(X))) |
(108) |
could be deleted.
1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[cons(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[recip(x1)] |
= |
1 · x1
|
[half(x1)] |
= |
1 · x1
|
[mark#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
mark#(cons(X1,X2)) |
→ |
mark#(X1) |
(85) |
|
1 |
> |
1 |
mark#(recip(X)) |
→ |
mark#(X) |
(88) |
|
1 |
> |
1 |
mark#(half(X)) |
→ |
mark#(X) |
(110) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
2nd
component contains the
pair
terms#(active(X)) |
→ |
terms#(X) |
(112) |
terms#(mark(X)) |
→ |
terms#(X) |
(111) |
1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[terms#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.2.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
terms#(active(X)) |
→ |
terms#(X) |
(112) |
|
1 |
> |
1 |
terms#(mark(X)) |
→ |
terms#(X) |
(111) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
cons#(X1,mark(X2)) |
→ |
cons#(X1,X2) |
(114) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(113) |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(115) |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(116) |
1.1.3 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[cons#(x1, x2)] |
= |
1 · x1 + 1 · x2
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.3.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
cons#(X1,mark(X2)) |
→ |
cons#(X1,X2) |
(114) |
|
1 |
≥ |
1 |
2 |
> |
2 |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(113) |
|
1 |
> |
1 |
2 |
≥ |
2 |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(115) |
|
1 |
> |
1 |
2 |
≥ |
2 |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(116) |
|
1 |
≥ |
1 |
2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
recip#(active(X)) |
→ |
recip#(X) |
(118) |
recip#(mark(X)) |
→ |
recip#(X) |
(117) |
1.1.4 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[recip#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.4.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
recip#(active(X)) |
→ |
recip#(X) |
(118) |
|
1 |
> |
1 |
recip#(mark(X)) |
→ |
recip#(X) |
(117) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
5th
component contains the
pair
sqr#(active(X)) |
→ |
sqr#(X) |
(120) |
sqr#(mark(X)) |
→ |
sqr#(X) |
(119) |
1.1.5 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[sqr#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.5.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
sqr#(active(X)) |
→ |
sqr#(X) |
(120) |
|
1 |
> |
1 |
sqr#(mark(X)) |
→ |
sqr#(X) |
(119) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
6th
component contains the
pair
s#(active(X)) |
→ |
s#(X) |
(122) |
s#(mark(X)) |
→ |
s#(X) |
(121) |
1.1.6 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[s#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.6.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
s#(active(X)) |
→ |
s#(X) |
(122) |
|
1 |
> |
1 |
s#(mark(X)) |
→ |
s#(X) |
(121) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
7th
component contains the
pair
add#(X1,mark(X2)) |
→ |
add#(X1,X2) |
(124) |
add#(mark(X1),X2) |
→ |
add#(X1,X2) |
(123) |
add#(active(X1),X2) |
→ |
add#(X1,X2) |
(125) |
add#(X1,active(X2)) |
→ |
add#(X1,X2) |
(126) |
1.1.7 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[add#(x1, x2)] |
= |
1 · x1 + 1 · x2
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.7.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
add#(X1,mark(X2)) |
→ |
add#(X1,X2) |
(124) |
|
1 |
≥ |
1 |
2 |
> |
2 |
add#(mark(X1),X2) |
→ |
add#(X1,X2) |
(123) |
|
1 |
> |
1 |
2 |
≥ |
2 |
add#(active(X1),X2) |
→ |
add#(X1,X2) |
(125) |
|
1 |
> |
1 |
2 |
≥ |
2 |
add#(X1,active(X2)) |
→ |
add#(X1,X2) |
(126) |
|
1 |
≥ |
1 |
2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
8th
component contains the
pair
dbl#(active(X)) |
→ |
dbl#(X) |
(128) |
dbl#(mark(X)) |
→ |
dbl#(X) |
(127) |
1.1.8 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[dbl#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.8.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
dbl#(active(X)) |
→ |
dbl#(X) |
(128) |
|
1 |
> |
1 |
dbl#(mark(X)) |
→ |
dbl#(X) |
(127) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
9th
component contains the
pair
first#(X1,mark(X2)) |
→ |
first#(X1,X2) |
(130) |
first#(mark(X1),X2) |
→ |
first#(X1,X2) |
(129) |
first#(active(X1),X2) |
→ |
first#(X1,X2) |
(131) |
first#(X1,active(X2)) |
→ |
first#(X1,X2) |
(132) |
1.1.9 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[first#(x1, x2)] |
= |
1 · x1 + 1 · x2
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.9.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
first#(X1,mark(X2)) |
→ |
first#(X1,X2) |
(130) |
|
1 |
≥ |
1 |
2 |
> |
2 |
first#(mark(X1),X2) |
→ |
first#(X1,X2) |
(129) |
|
1 |
> |
1 |
2 |
≥ |
2 |
first#(active(X1),X2) |
→ |
first#(X1,X2) |
(131) |
|
1 |
> |
1 |
2 |
≥ |
2 |
first#(X1,active(X2)) |
→ |
first#(X1,X2) |
(132) |
|
1 |
≥ |
1 |
2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
10th
component contains the
pair
half#(active(X)) |
→ |
half#(X) |
(134) |
half#(mark(X)) |
→ |
half#(X) |
(133) |
1.1.10 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[half#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.10.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
half#(active(X)) |
→ |
half#(X) |
(134) |
|
1 |
> |
1 |
half#(mark(X)) |
→ |
half#(X) |
(133) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.